A greedy randomized algorithm achieving sublinear optimality gap in a dynamic packing model

We consider an infinite-server system, where multiple customers can be placed into one server for service simultaneously, subject to packing constraints. Customers are placed for service immediately upon arrival, and leave the system after service completion. A key system objective is to minimize the total number of occupied servers in steady state. We consider the GreedyRandom (GRAND) algorithm, introduced in [23]. This is an extremely simple customer placement algorithm, which is also easy to implement. In [23], we showed that a version of the GRAND algorithm - so called GRAND(aZ), where Z is the current total number of customers in the system, and a > 0 is an algorithm parameter - is weakly asymptotically optimal, in the following sense: the steady-state optimality gap grows as c(a)r, where r is the system scale, i.e., the expected total number of customers in steady state, and c(a) > 0 depends on a, and c(a) → 0 as a → 0. In this paper, we consider the GRAND(Zp) algorithm, where p ∈ [0, 1) is an algorithm parameter. We prove that for any fixed p sufficiently close to 1, GRAND(Zp) is strongly asymptotically optimal, in the sense that the optimality gap is o(r) as r → ∞, sublinear in the system scale r. This is a stronger result than the weak asymptotically optimality of GRAND(aZ).